Mathematical Tools for Physics

2—Infinite Series 36

The only difference between the infinite series on the left and on the right is one term, so either everything
converges or everything diverges.
You can do better than this and use these inequalities to get a quick estimate of the sum of a series that
would be too tedious to sum by itself. For example

∑∞

1

1

n^2

= 1 +

1

22

+

1

32

+

∑∞

4

1

n^2

This last sum lies between two integrals.

∫∞

3

dx

1

x^2

>

∑∞

4

1

n^2

>

∫∞

4

dx

1

x^2

(9)

that is, between 1/3 and 1/4. Now I’ll estimate the whole sum by adding the first three terms explicitly and
taking the arithmetic average of these two bounds.

∑∞

1

1

n^2

≈1 +

1

22

+

1

32

+

1

2

(

1

3

+

1

4

)

= 1. 653

The exact sum is more nearly 1.644934066848226, but if you use brute-force addition to achieve accuracy
equivalent to this 1.653 estimation you will need to take about 120 terms. This series converges, but not
very fast.

Quicker Comparison Test
There is another way to handle the comparison test that works very easily and quickly (if it’s applicable). Look
at the terms of the series for largenand see what the approximate behavior of thenthterm is. That provides a
comparison series. This is better shown by an example:

∑∞

1

n^3 − 2 n+ 1/n

5 n^5 + sinn